The Mathematics of Financial Modelingand Investment Management

2-Financial Markets Page 69 Wednesday, February 4, 2004 1:15 PM

Overview of Financial Markets, Financial Assets, and Market Participants 69

Call option price = ≥ Max (0, Price of asset – Strike price)

This expression says that the call option price will be greater than or

equal to the difference between the price of the underlying asset and the

strike price (intrinsic value), or zero, whichever is higher.

The boundary conditions can be “tightened” by using arbitrage argu-

ments coupled with certain assumptions about the cash distribution of the

asset.^10 The extreme case is an option pricing model that uses a set of

assumptions to derive a single theoretical price, rather than a range. Deriv-

ing a theoretical option price is much more complicated than deriving a

theoretical futures price, because the option price depends on the expected

return volatility of the underlying asset over the life of the option.

Several models have been developed to determine the theoretical

value of an option. The most popular one was developed by Fischer

Black and Myron Scholes in 1973 for valuing European call options.^11

Several modifications to their model have followed since then. We shall

discuss the Black-Scholes model and its assumptions in Chapter 15.

Basically, the idea behind the arbitrage argument is that if the payoff

from owning a call option can be replicated by purchasing the asset

underlying the call option and borrowing funds, the price of the option

is then (at most) the cost of creating the replicating strategy.

SWAPS

A swap is an agreement whereby two parties (called counterparties)

agree to exchange periodic payments. The dollar amount of the pay-

ments exchanged is based on some predetermined dollar principal,

which is called the notional principal amount or notional amount. The

dollar amount each counterparty pays to the other is the agreed-upon

periodic rate times the notional principal amount. The only dollars that

are exchanged between the parties are the agreed-upon payments, not

the notional principal amount. In a swap, there is the risk that one of

the parties will fail to meet its obligation to make payments (default).

This is referred to as counterparty risk.

Swaps are classified based on the characteristics of the swap payments.

There are four types of swaps: interest rate swaps, interest rate-equity

swaps, equity swaps, and currency swaps. In an interest rate swap, the

(^10) See Chapter 4 in John C. Cox and Mark Rubinstein, Option Markets (Englewood
Cliffs, N.J.: Prentice Hall, 1985), Chapter 4.
(^11) Fischer Black and Myron Scholes, “The Pricing of Corporate Liabilities,” Journal
of Political Economy (May–June 1973), pp. 637–659.